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Inventing conjectures which I can't ever possibly prove
is a strange activity, so why am I doing it? Well, back in 1994 I was more enthusiastic, that't one of the reasons.
Anyway, here is a conjecture: let F be a field. Let K be the field obtained by adjoining to F all roots of all orders of all elements of F. Then K is a field of homological dimension 1, that is, all Sylow subgroups of the absolute Galois group of K are free pro-l-groups.
Why do I believe it? Well, it is true for number fields (where it suffices to adjoin the roots of unity). And if F is Henzelian with respect to a discrete valuation and f is the residue field, then the conjecture is true for F whenever it holds for f. And this is about all the supporting evidence I have.
Anyway, here is a conjecture: let F be a field. Let K be the field obtained by adjoining to F all roots of all orders of all elements of F. Then K is a field of homological dimension 1, that is, all Sylow subgroups of the absolute Galois group of K are free pro-l-groups.
Why do I believe it? Well, it is true for number fields (where it suffices to adjoin the roots of unity). And if F is Henzelian with respect to a discrete valuation and f is the residue field, then the conjecture is true for F whenever it holds for f. And this is about all the supporting evidence I have.
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here is why
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